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The purpose of this thesis is to demonstrate the use of boundary element techniques for the dynamic analyses of geometrically repetitive structures using the traveling wave approach. A formulation of the boundary element method (BEM) for 2-D isotropic materials is developed. The BEM formulation is then used to calculate the mass and stiffness matrices of one bay of a baseline structure. From the mass and stiffness a transfer matrix is developed for the bay. Using traveling wave theory, the transfer matrix is then used to identify the dynamical characteristics of a multiple bay structure. Results are compared to continuum theory.

We apply the Generalized Minimal Residual (GMRES) iterative equation solution technique to a set of full, unsymmetric matrix systems generated by a standard boundary element method. The test problems chosen produced well conditioned matrices. The GMRES technique, when used without preconditioning and with a sufficient number of trial vectors, solved the matrix system using as few as 23% of the operations required by a direct Gauss reduction. The class of partial LU decomposition preconditioners tested degraded the condition number of the matrices, and consequently did not reduce the GMRES solution time. In general, the GMRES technique does not appear to be of practical interest compared to the direct reduction unless other factors (availability of a good approximation to the final solution, etc.) intervene.

Boundary-element technique research performed to solve soil-structure interaction problems is the basis for this report. Specifically, it was designed to answer questionable situations for long hydraulic U-Lock structures erected on elastic soil. A review of earlier research that employed the Kelvin-type element directed the decision in favor of Flamant-type boundary element. Both constant and linear elements were developed and many sample problems were dealt with to illustrate the capabilities of these elements in the problem situations of soil-structure relationships. The Flamant-type boundary element proved to be convenient, gave better results for displacement problems using the constant element, offered satisfactory behavior description of soil-structure response, required considerably fewer elements than the conventional model, provided more accuracy in stress analysis of structure, and was confirmed as very beneficial for use in input data preparation. Research was also conducted on soil-structure interaction problems with the structure modeled by the finite-element method and the soil modeled by the boundary-element method. A model problem suggested by the AEWES was selected for study and solving. The emphasis was to develop a convenient and economical technique to couple the boundary- and finite-element methods of analysis. A computer code for coupling the two methods has been developed in FORTRAN. Keywords: Soil mechanics; Structural engineering; Mathematical analysis.

This thesis advances boundary element techniques to model thermal oxidation of silicon in two dimensions. At temperatures encountered in thermal oxidation, silicon dioxide flows viscoelastically. A reduced-dimension, generalized boundary element method for modeling such a problem has been developed. With a Laplace transform technique, a viscoelastic kernel function in derived from Kelvin's solution, which is the fundamental solution to linear elasticity. Constant-velocity loading is chosen to operate with a wide range of stress relaxation times. This scheme is capable of replacing boundary methods developed for slow viscous flow and elastic deformation. The oxidant diffusion problem is solved using a standard potential method for Laplace problems. Generated by oxide growth, stress affects both oxidant diffusion and oxide flow. In particular, it changes the diffusivity of oxidants and viscosity of oxide, rendering the diffusion and flow problems nonhomogeneous. Domain solutions are needed for both processes. A unified initial stress/built-in filed formulation has been developed to account for the nonlinear effects, using interior cells that are placed where stress is significant.

In this paper, we couple regularization techniques with the adaptive $hp$-version of the boundary element method ($hp$-BEM) for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions. As a model example we treat the delamination of composite structures with a contaminated interface layer. This problem has a weak formulation in terms of a nonsmooth variational inequality. The resulting hemivariational inequality (HVI) is first regularized and then, discretized by an adaptive $hp$-BEM. We give conditions for the uniqueness of the solution and provide an a-priori error estimate. Furthermore, we derive an a-posteriori error estimate for the nonsmooth variational problem based on a novel regularized mixed formulation, thus enabling $hp$-adaptivity. Various numerical experiments illustrate the behavior, strengths and weaknesses of the proposed high-order approximation scheme.

In this paper, we couple regularization techniques with the adaptive $hp$-version of the boundary element method ($hp$-BEM) for the efficient numerical solution of linear elastic problems with nonmonotone contact boundary conditions. As a model example we treat the delamination of composite structures with a contaminated interface layer. This problem has a weak formulation in terms of a nonsmooth variational inequality. The resulting hemivariational inequality (HVI) is first regularized and then, discretized by an adaptive $hp$-BEM. We give conditions for the uniqueness of the solution and provide an a-priori error estimate. Furthermore, we derive an a-posteriori error estimate for the nonsmooth variational problem based on a novel regularized mixed formulation, thus enabling $hp$-adaptivity. Various numerical experiments illustrate the behavior, strengths and weaknesses of the proposed high-order approximation scheme.